Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

Abstract

We provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations. Some of the criteria involve a comparison with the spectral radii of some associated linear operators. We apply our results to prove the existence of multiple nonzero radial solutions for some systems of elliptic boundary value problems subject to nonlocal boundary conditions. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.Mathematics Subject Classification. Primary 45G15, secondary 34B10, 35B07, 35J57, 47H30.

Solutions of perturbed Hammerstein integral equations with applications

“…Nonlocal BCs have been also studied in the the context of elliptic problems, we mention the papers [4,41,76,77,81,83].…”

“…Here we focus on the systems of BVPs where x ∈ R n , β 1 ,β 2 < 0, 0 < R 1 < R 0 < +∞, R η , R ξ ∈ (R 1 , R 0 ), that can be seen as a generalization of system (7.1) studied in [41] (the range ofβ 2 corrects the one of α 2 in [41]).…”

Non-Zero Radial Solutions for Elliptic Systems with Coupled Functional BCs in Exterior Domains

“…In the case of [a 1 , b 1 ] = [a 2 , b 2 ] the assumptions on the nonlinearities f i can be relaxed, for example in condition (I 0 ρ 1 ,ρ 2 ) • we have to control the growth of f i in the smaller set [0, ρ 1 /c 1 ] × [0, ρ 2 /c 2 ]. We refer to the paper [27] for the statement of similar results.…”

“…The case u = ρ 1 and v ≤ ρ 2 is simpler and therefore is omitted. Note that, using the inequality (2.11) as in [27,28], it is possible to relax the growth assumptions on the nonlinearity f 2 in Lemma 2.5, by replacing the quantity sup t∈[0,1]…”

Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria

“…In 2011, Lan and Lin studied, via spectral theory, the existence of positive solutions of systems of Hammerstein integral equations in [25,26] and Lan [24] proved a new result for the existence of positive solutions of systems of second order elliptic boundary value problems. In 2015, Infante and Pietramala in [17] established some criteria for the existence of solutions of systems of Hammerstein integral equations that involve a comparison with the spectral radii of some associated linear operators. The Krein-Rutman Theorem, that is a celebrated result of the spectral theory, requires the totality of the cone.…”